Math 115: Review for Chapter 2
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1 Math 5: Review for Chapter Can ou determine algebraicall whether an equation is smmetric with respect to the - ais, the -ais, or the origin?. Algebraicall determine whether each equation below is smmetric with respect to the - ais, the -ais, or the origin. a. b. c. = + = 0 = + 5 d. = + 9 e. = Can ou determine graphicall whether an equation is smmetric with respect to the - ais, the -ais, or the origin?. Using visual observation, determine whether each graph is smmetric with respect to (a) the ais, (b) ais, (c) origin. a. b. c. Can ou determine algebraicall whether an equation is even or odd?. Algebraicall determine whether each function below is even or odd. f = + a. ( ) f = b. ( ) 6 c. f ( ) = d. f ( ) = e. f ( ) = 6
2 . Use the curve of f below to answer the following questions. Assume that each bo measures unit b unit. a. What are f ( 0) and ( ) f? b. Is f ( ) positive or negative? c. What is the domain of f? d. For what values of is f increasing? decreasing? e. What are the and intercepts? f. For what values of does f ( ) =? g. Is the graph of f ( ) continuous? Eplain wh or wh not. h. For what values of is f constant? Given the curve of f ( ) below, can ou sketch the curve of the function ( ) where a, b, c, h, and k are constants? af b + h + k, 5. Graph g ( ) = + using a graphing utilit and then list all the transformations applied to the graph of =.
3 6. Given the graph of f ( ) below, sketch the graphs of the following functions. a. f ( + ) b. f ( ) c. f ( ) d. f ( ) e. + f ( ) f. f ( ) g. f ( ) 7. The graphs below each represent a transformation of the curve of each of the graphs. =. Write an equation 5 5 a. c. 5 5 b. d.
4 8. Starting with the graph of the basic function =, graph each of the following functions b hand using transformation techniques. a. = + b. = + c. =
5 9. The graph of f ( ) is given below. Use transformations to sketch the curve of the following functions. Be sure to label at least four points on the new curve a. g ( ) = f ( ) + c. k ( ) = f ( ) b. h ( ) = f ( ) d. l ( ) = f ( )
6 Can ou graph and recognize some of the basic functions? Basic Functions Linear Function f ( ) = m + b Constant Function f ( ) = b Identit Function f ( ) = Square Function f ( ) = Cube Function f ( ) = Square Root Function f ( ) = Reciprocal Function f ( ) = f = int = Greatest Integer Function ( ) ( ) 0. Sketch the graph of each of the following functions. a. ( ) f = b. ( ) g = c. h ( ) =. If f ( ) = int ( + ), find each. a. f (.7) b. f (.7) c. f (.) Given a piecewise-defined function, can ou evaluate the function at different values of and then sketch a graph of the function?. Given ( ) if < f = if =, + if > a. find f ( 0), f ( ), f ( 5), ( ) b. find the domain of f; c. sketch a graph of f; and d. Use the graph to find the range of f. f, and f ( ) + ;
7 . Write a definition of the function whose graph is given below.. What is the definition of the function whose graph is given below? Can ou solve absolute value equations? 5. Solve each absolute value equation. Find real solutions. a. t + = 6 e. 5 = 0 b. t + 6 = c = 7 f. g. = = d. = 5 h. 7 7 =
8 Can ou solve absolute value inequalities? 6. Solve each absolute value inequalit. Use interval notation to write the solution. a. < 7 e. + 7 > b. + 8 f c. 5 7 g. d. 9 7 < h. 9 7 > 7. If, find a and b so that a + b. Given two functions, can ou find their sum, difference, product, and quotient? 8. Given f ( ) = and ( ) = +, find each. g a. ( f + g) ( ) b. ( f + g) ( ) c. ( f g) ( ) f g e. ( ) f. the domain of f g d. ( f g) ( ) 9. Given f ( ) = and g ( ) =, find each. f + g (a) ( ) (b) ( f + g) ( ) (c) ( f g) ( ) (d) the domain of f + g
9 Given two functions f and g, can ou find the composite function f o g or go f? 0. Given f ( ) = and g( ) =, find each. + a. ( f o g)( ) b. ( go f)( ) c. ( f o f)( ) d. the domain of go f. Given h( ) = 5 and k( ) =, find each. a. ( ho h)( ) c. ( ko k)( 0) b. ( ho k)( ) d. ( ho k)( ). Find two function f and g such that ( f g)( ) = k( ) = 5 o.. Given f ( ) = and g( ) + =, show that ( )( ) ( )( ) f o g = go f =.
10 . Given the graphs of f ( ) and ( ) f ( ) g( ) g below, answer the following questions: 5 5 a. Find ( g f)( ) b. Find ( f o g)( ) c. Find ( go g)( ) 5 5 f g d. Find ( ) e. Find ( f + g)( )
11 Answers:. a. smmetric with respect to the origin b. smmetric with respect to the -ais c. not smmetric to an of the -ais, -ais, or origin d. smmetric with respect to the origin e. smmetric with respect to the -ais. a. smmetric with respect to the origin b. smmetric with respect to the ais, ais, or origin c. smmetric with respect to the origin. a. Even b. Odd c. Neither d. Even e. Even. a..5; 0.75 b. positive c. [ 8,8] d. increasing: ( 8, 5.5) (,) ; decreasing: ( 5.5, ) (,8) e. -intercepts: -8, -.5, -0.75, and 5; -intercept:.5 f. = { 7.5,., 0.,.} g. Yes. There s no jumps, holes, or breaks in the graph h. f is never constant 5. Reflect the basic over the -ais and then shift the reflected curve four units to the right. 6. (a) Shift the given curve unit to the left. (b) Shift the given curve unit to the right. (c) Verticall stretch the given curve. (d) Reflect the given curve about the -ais. (e) Move the given curve up units. (f) Reflect the given curve about the -ais. (g) Verticall stretch the given curve, then shift units down. 7. (a) f ( ) = + (b) f ( ) = ( + ) (c) f ( ) = ( ) + (d) f ( ) ( ) = + 8. (a) Shift the basic graph of = three units to the left. (b) Shift the basic graph of = one unit to the left and then move the entire graph units down. (c) =. Graph =. Reflect the basic graph of about the -ais. 9. (a) Move the given curve up units. (b) Shift the given curve units to the right. (c) Reflect the given curve about the -ais. (d) Shrink the basic curve horizontall. 0. See sketches of the graphs in our tetbook.. (a) (b) (c)
12 . (a) f ( 0) = ; f ( ) = ; f ( 5) = ; ( ) ( ) ( ) f + = + + f = 7; (b) all real numbers (d) all real numbers. f ( ) if < = if. f ( ) = 5. a. t = or t = b. no solution c. = or = 0 7 d. = or = 0 0 e. = ± 5 f. = or = g. = 5 or = or = or = 0 h. = or = 6. (a) < < 5 ; (,5) (b) (c) ;, (d) (e) < 5 or, 5, (f) or 5 (g) or > ; ( ) ( ) 5,, ;, [, ) ; ( ] [ ) (h) all real numbers; (, ) 7 5 ; no solution 7 5, 7. a = and b = 5 8. a. + b. c. d. 95 e. f. all real numbers + 9. a. b. + c. d. { 0 } 0. a. 5 b. c. - d. all real numbers ecept + +. a. -8 b. -0 c. - d. 8. one eample: f ( ) =, g ( ) = 5. simplif both ( g o f) ( ) and ( f o g) ( ). a. b. c. d. / e. 6 Revised 9//07
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